File location naming hierarchy

ABSTRACT

Embodiments of methods, apparatuses, devices and/or systems for a file location naming hierarchy are disclosed.

This disclosure claims priority pursuant to 35 USC 119(e) from U.S. provisional patent application Ser. No. 60/584,734, filed on Jun. 30, 2004, by J. J. LeTourneau, titled, “FILE LOCATION NAMING HIERARCHY,” assigned to the assignee of the presently claimed subject matter.

BACKGROUND

This disclosure is related techniques for naming file locations, such as for a file location naming hierarchy.

In a variety of fields, data or a set of data, may be represented in a hierarchical fashion. This form of representation may, for example, convey information, such as particular relationships or patterns between particular pieces of data or groups of data and the like. However, manipulating and/or even recognizing specific data representations or patterns is not straight-forward, particularly where the data is arranged in a complex hierarchy. Without loss of generality, one example may include a database, such as a relationship database. Typically, a relational database includes files arranging in a file naming hierarchy that provides location information so that the file may be accessed. Techniques for manipulating files, defining relationships among files and/or moving between files in such a hierarchy continue to be desirable.

BRIEF DESCRIPTION OF THE DRAWINGS

Subject matter is particularly pointed out and distinctly claimed in the concluding portion of the specification. The claimed subject matter, however, both as to organization and method of operation, together with objects, features, and advantages thereof, may best be understood by reference of the following detailed description when read with the accompanying drawings in which:

FIG. 1 is a tree diagram illustrating an embodiment of a file location naming hierarchy;

FIG. 2 is a table illustrating an embodiment of names and relationships for an embodiment of a file location naming hierarchy;

FIG. 3 is a table illustrating another embodiment of names and relationships for an embodiment of a file location naming hierarchy;

FIG. 4 is a schematic diagram illustrating another embodiment of a file location naming hierarchy;

FIG. 5 is a schematic diagram illustrating yet another embodiment of a file location naming hierarchy.

DETAILED DESCRIPTION

In the following detailed description, numerous specific details are set forth to provide a thorough understanding of the claimed subject matter. However, it will be understood by those skilled in the art that the claimed subject matter may be practiced without these specific details. In other instances, well-known methods, procedures, components and/or circuits have not been described in detail so as not to obscure the claimed subject matter.

Some portions of the detailed description which follow are presented in terms of algorithms and/or symbolic representations of operations on data bits or binary digital signals stored within a computing system memory, such as a computer memory. These algorithmic descriptions and/or representations are the techniques used by those of ordinary skill in the data processing arts to convey the substance of their work to others skilled in the art: An algorithm is here, and generally, considered to be a self-consistent sequence of operations and/or similar processing leading to a desired result. The operations and/or processing involve physical manipulations of physical quantities. Typically, although not necessarily, these quantities may take the form of electrical and/or magnetic signals capable of being stored, transferred, combined, compared and/or otherwise manipulated. It has proven convenient, at times, principally for reasons of common usage, to refer to these signals as bits, data, values, elements, symbols, characters, terms, numbers, numerals and/or the like. It should be understood, however, that all of these and similar terms are to be associated with the appropriate physical quantities and are merely convenient labels. Unless specifically stated otherwise, as apparent from the following discussion, it is appreciated that throughout this specification discussions utilizing terms such as “processing”, “computing”, “calculating”, “determining” and/or the like refer to the actions and/or processes of a computing platform, such as a computer or a similar electronic computing device, that manipulates and/or transforms data represented as physical electronic and/or magnetic quantities and/or other physical quantities within the computing platform's memories, registers, and/or other information storage, transmission, and/or display devices.

In a variety of fields, data or a set of data, may be represented in a hierarchical fashion. This form of representation may, for example, convey information, such as particular relationships or patterns between particular pieces of data or groups of data and the like. However, manipulating and/or even recognizing specific data representations or patterns is not straight-forward, particularly where the data is arranged in a complex hierarchy. Without loss of generality, one example may include a database, such as a relationship database. Typically, a relational database Includes files arranging in a file naming hierarchy that provides location information so that the file may be accessed. Techniques for manipulating files, defining relationships among files and/or moving between files in such a hierarchy continue to be desirable. In this context, the term file refers to any stored collection of data that is segregated or partitioned in some manner, such as, for example, audio data, video data, computer code, text, etc.

In this context, a location naming hierarchy refers to a technique or method of naming a location in which data is stored or to be stored, including, but not necessarily limited to including, a collection of operations to be performed upon those named locations that, if performed upon the location names, performs operations on the data stored in those named locations to produce another set of data that may be stored in another named location. One example of a location naming hierarchy commonly used is the “Unix” file hierarchy. As is well-known, in the Unix hierarchy, a particular file may have a name for an associated location that comprises a collection of finite character sequences, such as, for example, the file name “/A/B/C/***/Z”, in which, for this example, A,B, C, etc. comprise file names in the form of finite sequences. For the purposes of simplification of discussion, it is appropriate to think of a sequence of characters as a finite sequence of natural numerals. Thus, for example, a technique, such as a look up table, may be employed to convert between finite sequences of natural numerals and finite sequences of characters. This may be employed, for example, in a computer or computing system in which the computer or computing system performs the conversion. Likewise, in addition, the natural numerals may likewise be converted to binary numerals so that they may be operated upon by the computer or computing system, although this level of detail is well-understood and need not be discussed further. Likewise, for the purposes of this discussion, the character “/” may be replaced by a comma and, therefore, the file name above, for example, may be represented as the following character string <A,B,C,***Z,>. These changes in nomenclature are primarily for purposes of convenience and to simplify the discussion. One of ordinary skilled in the art will appreciate that there is no loss of generality by making these substitutions.

With this understanding, reference is now made to FIG. 1. FIG. 1 depicts a tree intended to convey, in graphical form, a potential embodiment of a location naming hierarchy (hereinafter referred to as “hierarchy”), such as one that may be employed in a Unix-based system. In this context, the term “Unix” or “Unix-based system” generally refers to a computing system or a computer system in which the Unix operating system or a version of the Unix operating system is employed. Of course, it is likewise noted that the file structure described above is also common in other operating systems other than a Unix based system, such as where other characters separate finite sequences, etc.

Referring now to FIG. 1, as illustrated, the tree includes a root node, designated 110 in FIG. 1. For the purposes of this discussion, the root node is referred to as being at “level 0”. Thus, here, level 0 has a single node, the root node. In FIG. 1, directly emanating from the root node are four nodes on a level referred to in this context as “level 1.” For the purposes of designation in a Unix-based operating system, from the prior discussion, the nodes on this level may be designated respectively as 1, 2, 3, and 4. Thus, in FIG. 1, node 1 has reference numeral 120, node 2 has reference 130, node three has reference numeral 140, and node 4 has reference numeral 150. Continuing with this approach, for this example, there is a level 2, a level 3, and a level 4. The level 2 includes five nodes, which may be referred to in a Unix-based operating system as 1, 2, 3, 4, and 5, respectively. The level 3 includes four nodes referred to here as 1, 2, 3, and 4, respectively, and the level 4 includes five nodes referred to here as 1, 2, 3, 4, and 5, respectively.

As illustrated in FIG. 1, therefore, a file represented in the tree as a node, using the nomenclature previously described, may be stored in a location named <2, 4, 3 1>, as shown in FIG. 1 with reference numeral 160. Likewise, the node that is, in this example, the file with the reference numeral 170 has a standard Unix-based operating system designation or location name of <4, 1, 1>. Thus, as is clear, from the prior discussion, each of these nodes, in this example, represents a particular file having designations or named locations in memory typical of the Unix-based operating system. Tree 100 in FIG. 1, therefore, is a graphical illustration of the file hierarchy for this particular embodiment.

Although the scope of the claimed subject matter is not limited in the following respect, a significant portion of data that is typically stored in connection with a computer system or a computing system is stored on a device or on devices conventionally referred to as a “hard drive” or on some other storage medium in which the data is typically accessed in a serial fashion. This is in contrast to memory conventionally referred to as Random Access Memory or RAM, in which the data may typically be accessed in a non-serial fashion. Thus, for a memory system in which data is accessed serially, such as a hard drive, as an example, accessing files is typically more time consuming than accessing data using RAM or Random Access Memory. A hard drive, for example, typically locates the position of a desired file on the disk of the hard drive to access the particular file. Thus, accessing a file involves finding the location of the particular file and then aligning the file location with a device that is capable of reading the stored data from that location. In comparison, for accessing data stored in RAM, the locations are, in essence, substantially simultaneously or equivalently accessible. Thus, accessing data from RAM typically takes less time at least in part because it does not involve locating the file and aligning that location with a device capable of reading the stored data. All or most locations in RAM are typically capable of being read. Thus, a file may be accessed once it is known which one is desired.

An aspect of knowing the location of a particular named file on a disk, for example, of a computing system employing a Unix-based operating system, involves knowing the positional relationship of a particular file to other files stored on the disk. Thus, for example, If a particular program is executing or operating in which, initially, the file named or designated in FIG. 1 as <2, 4, 3, 1> is accessed and, in succession, the file named or designated as <4, 1, 1> is accessed, the operating system, in essence, traces file locations based at least in part on the naming convention employed, in this example by moving up the particular file hierarchy in memory, here along a branch from the bottom node, designated by reference 160 in FIG. 1, back to the root node designated 110 in FIG. 1, and then along another branch from root node 110 to node 170. In other words, in a Unix-based operating system, the “linear” relationship of the names of the particular files at each of the levels of the hierarchy or structure provides a mechanism to move between files in memory based at least in part on the particular hierarchy. This particular approach to finding the location of named files to be accessed serially may be time consuming. Of course, in this particular context, the frame of reference for the term “time consuming” may change as technology advances. For example, in one frame of reference, time consuming may refer a relatively short time in comparison to seconds, but a relatively long timing in comparison to the speed of, for example, a state-of-the-art RAM. However, again, such frames of reference may be subject to change.

It is likewise noted that an alternative approach to the one just described may include maintaining the positional relationship between every file to every other file. A disadvantage, however, to this approach is that as the number of files increase, maintaining and using this amount of information effectively may become intractable.

An embodiment of a method of arranging files in file hierarchy will now be discussed in which the relationship among files is such that moving between files in the hierarchy is typically less time consuming than the approach just discussed above in connection with a Unix-based operating system. In this particular embodiment, the file structure or hierarchy is based, at least in part, on a location naming convention described in this context as employing finite multi-sets of natural numerals. In comparison, as alluded to by previously, the file structure or hierarchy employed in a Unix-based operating system is based, at least in part, on a location naming convention described in this context as employing finite sequences of natural numerals. As shall be explained in more detail below, in this particular context, a finite sequence refers to a collection of a finite number of elements indexed by the natural numbers. Likewise, in this particular context, a finite multi-set refers to an unordered collection of a finite number of elements. It is noted that the term collection includes situations in which elements are repeated. Thus, whereas for a finite sequence, the order or indexing of the arrangement of elements, for this embodiment, here, natural numerals, is an aspect of the naming convention, as well as the multiplicity or repetition of particular elements; in contrast, for a finite multi-set, the multiplicity or repetition is an aspect of the naming convention, but the order or indexing of the elements is not.

For this particular embodiment, FIG. 2 shows a table containing four columns; however, the ellipse in FIG. 2 is intended here to indicate a table of an unspecified number of total columns. As is apparent from FIG. 2, the first column contains the natural numerals in an ascending order. Likewise, as shall be explained in more detail hereinafter, the second column contains the finite multi-sets of the natural numerals, each finite multi-set set corresponding to a natural numeral for this particular embodiment. More specifically, using, for this embodiment, particular rules or principles of construction, the second column has, as its elements, all possible finite multi-sets created based on the natural numerals. These finite multi-sets corresponding to natural numerals, here, are referred to in this context as finite multi-sets or FMS, for convenience.

To construct the second column, for this particular embodiment, a specific technique is employed to construct the multi-set of the natural numerals. Of course, the claimed subject matter is not limited in scope to this particular embodiment and, in particular, is not limited in scope to this particular technique or these rules of construction. Thus, other rules or other approaches to constructing or creating a file hierarchy naming convention using a multi-set of the natural numerals, for example, may be employed and still remain within the scope of the claimed subject matter. Nonetheless, as suggested above, for this particular embodiment, a particular finite multi-set is associated with each natural numeral. As will be demonstrated, by repetitively applying these rules of construction, multi-sets of multi-sets shall be created, as explained in more detail hereinafter.

Thus, for the natural numeral, zero, for example, the empty set { } is associated, for this embodiment, as illustrated in column two, along the row that contains zero in the first column. Likewise, for the natural numeral, 1 of column one, the set {0} of column two is associated, which may also be referred to here as “singleton zero.” For the natural numeral, 2, the finite multi-sets {0,0} is associated, which may be referred to here as “doubleton zero.” Thus, in this particular embodiment, although the claimed subject matter is not limited in scope in this respect, the natural numeral zero is associated with the simplest multi-set. The natural number one is associated with the next simplest multi-set, etc.

It is noted that the natural numeral 2 is the only even non-composite numeral for this particular embodiment. An approach or rule for constructing a finite multi-set associated with an odd non-composite numerals is as follows, for this embodiment. For any natural numeral k, in which k is an odd non-composite numeral, a singleton set {i} is associated with it, in which i is the index of the non-composite numeral in the well-known Kleene enumeration. For example, the set associated with natural numeral 3 is {1}. The set associated with natural numeral 5 is {2}. The set associated with natural numeral 7 is {3}, etc.

The remaining rules refer to forming a finite mull-set associated with a natural numeral from the prior finite multi-sets associated with the natural numerals. For example, any even composite natural numeral may be expressed as 2 k, in which k is a prior natural numeral such that an associated finite multi-set has been defined, such as above. The finite multi-set of 2 k, here is formed from a union of the finite multi-set associated with k and the finite multi-set {0}. Likewise, any odd composite numeral, k*L, has associated with it the multi-set that is formed from the union of the multi-sets associated with k and L, respectively, for this embodiment. Thus, for this particular embodiment, although the claimed subject matter is not limited in scope in this respect, merger of the index of non-composite natural numerals corresponds to conventional multiplication.

Returning to FIG. 2, the set associated with the natural numeral 4 is a set that includes three zeros {0,0,0}—the merger of the set associated with 2 with the set {0}, as explained above. Likewise, the set associated with the natural numeral 8 comprises four zeros {0,0,0,0} and so forth. Furthermore, the set associated with 6 is the merger of the set associated with 3, here {1}, with the set that contains one zero {0}.

Continuing to column three of FIG. 2, its' elements comprise the finite multi-sets of column two. Column three, here, is constructed by applying the rules described above to column two. Likewise, the rules applied to column three construct column four, etc. It is also noted that an alternate way to move from one column to another column is to substitute the natural numerals in the column with their associated set from column two. Thus, for example, for natural numeral 7, its element in the third column may be constructed from the second column by replacing 3, which is along the same row, with its associated set {1}. Likewise to move from column three to column four along the same row, the natural numeral 1 in {1} is replaced with its associated set {0} to produce {{0}} and so forth.

Although the claimed subject matter is not limited in scope in this respect, this ability to move between columns and, hence, between multi-sets, may provide advantages in certain environments. It allows the potential for inter-related file structures. For example, in some situations, it may be desirable to run multiple tasks using related file structures. Computer science has many examples where this may occur, such as with databases, computer programs, look ahead operations, etc. Thus, an embodiment in which the file structures employs an approach similar to that just described, for example, provides the ability to move between files created as part of performing such disparate but related tasks, quickly and easily. Depending upon the particular situation, it may simply comprise moving between columns in a file structure, such as the file structure embodiment above, for example.

The previously described approach for constructing multi-sets far the natural numerals, here column two, provides a technique for associating a natural numeral, here column one, with a finite multi-set in which the elements of the set are the natural numerals. It may be convenient to view a finite multi-set as a string for the purposes of implementation on a computing device, such as a computer, for example. As will be explained in more detail hereinafter, this association may be usefully employed in a variety of contexts, such as, for this embodiment, to construct a file hierarchy, for example.

In this particular embodiment in which a table, such as that illustrated in FIG. 2, is employed to create a file hierarchy, file names are employed that comprise multi-sets. In this particular embodiment, the multi-set is related to the number of columns to be constructed. In this context, the term “multi-set depth” indicates that the file names are taken from a particular column such as if the columns are constructed as previously described, up to a particular column. For example, if the Multi-set depth is 6, ultimately, for this embodiment, it will be desirable to construct the seventh column for this particular embodiment, as explained in more detail hereinafter. As may be observed, for this embodiment, column two in FIG. 2 has a multi-set depth of 1. For this example, therefore, it is desirable to construct columns corresponding to depths of 6, 5, 4, 3, 2 and 1, and also to construct a K+1 or seventh column that corresponds to the multi-set depth of K, here K being 6 to correspond to a multi-set depth of 6. By constructing a column of multi-set depth 6, here column 7, a column has been created in which the file names employed in this column is related to the file names in the other prior columns along a corresponding row, for example, although, of course, the claimed subject matter is not limited in scope in this respect.

By this construction, the multi-set file names desired to be employed may be related to natural numerals by moving between the K+1 column and the first column. This, therefore, permits movement between hierarchical file names in a process that may be implemented, for example, by, for this embodiment, moving between columns and up and down rows. Such an approach, thus, may be more efficient than the approach previously described in connection with the Unix-based operating system at least in part because it permits direct movement between two files rather than tracing branches of a tree, as previously described. It is likewise noted that the multi-set depth, or the column from which a particular file name was selected, may be determined “on the fly,” in this embodiment, by the number of sets of parentheses in the file name. Of course, the claimed subject matter is not limited to employing parentheses and, thus, other techniques to make such a determination on the fly are also possible and included within the scope of the claimed subject matter. However, as noted, for a particular file name, its row and column location may be determined from the name so that the operations described above may be quickly and easily implemented if such a file structure or a similar file structure is employed.

Of course, techniques for performing table look-ups are well-known and well-understood. Thus, this will not be discussed in detail here. However, it shall be appreciated that any and all of the previously described and/or later described processing, operations, conversions, transformations, manipulations, etc. of strings, trees, numerals, data, etc. may be performed on one or more computing platforms or similar computing devices, such as those that may include a memory to store a table as just described, although, the claimed subject matter is not necessarily limited in scope to this particular approach. Thus, for example, a hierarchy of data, such as a tree as previously described, for example, may be formed. Likewise, operations and/or manipulations, as described, may be performed; however, operations and/or manipulations in addition to those described or instead of those described may also be applied. It is intended that the claimed subject matter cover all such embodiments.

For this particular embodiment, although the claimed subject matter is not limited in scope in this respect, the files may be arranged in a naming hierarchy in which the hierarchy is constructed by recursively forming multi-sets of a set of elements. In this particular embodiment, for example, the set of elements may comprise the finite multi-set of natural numerals, although, again the claimed subject matter is not limited in scope in this respect. Recursively forming of multi-sets may, therefore, create a naming hierarchy for files, such as just described for this particular embodiment. In this context, recursively forming multi-sets may also be referred to as compound multi-sets.

FIG. 3 illustrates another embodiment of a table in accordance with the claimed subject matter. As is apparent from inspection, the first two columns of the table of FIG. 2, correspond to the first two columns of the table of FIG. 3, although, again, the claimed subject matter is not limited in scope in this respect. Column one of FIG. 3, therefore, also contains natural numerals in ascending order and column two of FIG. 3 also contains corresponding finite multi-sets, as previously described. Column three of FIG. 3, however, for this embodiment, contains numeral pairs, as shall be described in more detail hereinafter.

In this particular embodiment, natural numerals of column one are related to numeral pairs of column three, as described in more detail below. One numeral of the numeral pair contains the index numeral for an associated non-composite numeral, such as previously described in connection with the Kleene enumeration. For example, as previously described, for the natural numeral 3, the index associated is 1. Likewise, for the natural numeral 5, the index associated is 2. For the natural numeral 7, the index associated is 3, etc.

The other numeral of the numeral pair, for this embodiment, comprises the numeral that produces the natural numeral of column one when multiplied by the non-composite numeral corresponding to the index of the Kleene enumeration. Again, using the natural numeral 3 of column one as an example, one numeral of the numeral pair is 1, as previously described. Thus, the other numeral of the numeral pair is also 1. More particularly, 1 times the non-composite numeral, for this example, 3, produces the natural numeral, again in this example 3. A more helpful may be the natural numeral 6 of column one. Here, the index may be either a 1 or a 0, as explained in more detail hereinafter. If the index is 1, the other natural numeral of the numeral pair is 2, that is, 3 (corresponding to 1) times 2 is 6. If, instead, the index is 0, the other numeral is 3 since 2 (corresponding to 0) times 3 is also 6.

As may now be appreciated, column three, in this embodiment, provides a mechanism to identify those natural numerals of column one that are non-composite. In particular, the first numeral is the index described previously, and the second numeral is the numeral that produces the natural numeral of column one, if multiplied by the non-composite numeral corresponding to the index. Thus, for this embodiment, numeral pairs in which the second numeral is 1 correspond to those natural numerals in column one that are non-composite. It is, again, appreciated that the claimed subject matter is not limited in scope to this particular embodiment. For example, the numerals may not be arranged in columns and/or the numeral pair may be reversed. Likewise, instead of a numeral pair in one column, perhaps two columns, each having a numeral may be employed. In the latter approach, for example, finding a one numeral in the column will identify the non-composite numerals. Many different approaches and/or embodiments may be employed; however, it is intended that they all be included within the scope of the claimed subject matter. For this embodiment, the numeral pairs in column three comprise a first operation in a factorization of the natural numeral in column one. Hence, for non-composite numerals, the identified numeral will be 1, whereas for other natural numerals that are not non-composite, this particular numeral will be something other than 1.

Column three, in this particular embodiment, also provides a mechanism through which it is possible to represent a natural numeral using fewer bits than an associated conventional binary representation. Consider, for example, the natural numeral 9. To represent 9 in a binary representation employs four bits, as is well-known. However, the column three representation of the natural numeral 9 is the numeral pair [1,3], in this particular embodiment. To represent the natural numeral 1 employs one bit. To represent the natural numeral 3 employs two bits. Therefore, by substituting the pair [1,3] in memory for 9, for example, three bits may be employed to represent the natural numeral 9, whereas conventionally it takes four bits. As should now be apparent, it is possible to produce the natural numeral of column one with the corresponding numeral pair of column three and vice-versa. For example, using again the natural numeral 9, its corresponding numeral pair for this embodiment is [1,3]. Here, 1 is the index for natural numeral 3, as indicated previously. Thus, 3 (corresponding to one) times 3 provides 9, in this example.

It is noted that, for this embodiment, it is not the case that employing the numeral pairs of column three always save a bit. However, for all cases, the binary representation associated with the third column, again, for this embodiment, will be less than or equal to the number of bits conventionally employed to represent the particular natural numeral. For example, consider natural numeral 8 in column one. The column three representation is [0, 4]. Eight is conventionally represented using four bits. Zero is represented using one bit. Four is represented using three bits. Therefore, both representations employ the same number of bits in this example, here, four. Thus, using this approach ensures that the same or fewer bits than is conventionally the case will represent a particular natural numeral

Another feature of this particular embodiment, illustrated in FIG. 3, is the association of a unique finite multi-set with the natural numeral of column one and/or the numeral pair of column three. Consider, as previously discussed, producing the associated finite multi-set from the particular natural numeral. For this particular embodiment, it should also be possible to produce this finite multi-set from column three. Consider, for example, the case of natural numeral 105. The associated finite multi-set is {1,2,3}. Associated numerical pairs include: [1,35]; [2,21]; and [3,15]. Here, the first numeral of the numeral pair is an element of the finite multi-set for this particular embodiment. The second numeral for the numeral pair is the product of the non-composite numerals associated with the remaining elements of the multi-set. For example, the non-composite numeral associated with 2 is 5 and the non-composite numeral associated with 3 is 7. Thus, the product of 5 and 7 is 35.

Thus, for this particular embodiment, the table of FIG. 3 provides, for a particular natural numeral, the finite multi-sets of the natural numeral and the numeral pair of the natural numeral. Likewise, the, numeral pair of this embodiment is analogous to the Kleene enumeration extended to include composite numerals. In particular, using the fundamental principle of arithmetic, one may always break any natural numeral into a unique set of non-composite numerals. Table three makes use of this fundamental precept by replacing the non-composites with their associated indexes. Furthermore, as previously demonstrated, for any natural numeral, in this embodiment, there is an association between these indexes and the finite multi-sets of column two.

Another feature of this particular embodiment includes employing numeral pairs to perform multiplication. For example, natural numeral 3 is represented by the pair [1,1]. Likewise, natural numeral 5 is represented by pair [2,1], In this particular embodiment, a general rule for combining numeral pairs is as follows:

[a,b] combined with [c,d]=[a, b*(Q(c)*d)],   [1]

-   -   in which “*” represents multiplication and where the function         Q(c) is a function that provides the non-composite numeral         associated with the index c, as for the Kleene enumeration.         Thus, in this example, merging the pairs produces the ordered         pair [1,5]. This numeral pair corresponds to the natural numeral         15, which is a product of 3 times 5. Furthermore, since, as         previously demonstrated, column three provides a mechanism to         represent the natural numeral of column one with the same or         fewer bits, this also provides a mechanism to perform         multiplication with the same of fewer bits than conventionally         employed.

For this embodiment, the table of FIG. 3 provides a mechanism to manipulate finite multi-sets, or employing the language of computer science, to manipulate strings, using a table-driven mechanism. For example, multiplication, such as 3 times 5, combines the finite multi-setss corresponding to those natural numerals. This may be accomplished, for this embodiment, therefore, either by multiplying the natural numerals of column one or by using the merger operation of column three, as described above. Thus, by a table look up mechanism, strings may be combined, such as in the manner described above for this particular embodiment.

Referring now to FIG. 4, a schematic diagram illustrates an embodiment of a binary tree file hierarchy “rooted” upon a finite multi-sets. As shall be explained in more detail hereinafter, this particular embodiment of a file hierarchy may be constructed based at least in part upon the table of FIG. 3. Consider, for example, the natural numeral 12 of column one. In column two, this corresponds to the finite multi-sets {0,0,1}. Using the table of FIG. 3, for this embodiment, associated with this finite multi-sets, and thus; also associated with natural numeral 12, are two unique non-composite numerals. It is noted that there are many different unique pairs of non-composite numerals that may be associated with natural numeral 12. This is merely one embodiment and the claimed subject matter is not limited in scope to this particular embodiment. However, one non-composite numeral comprises Q(2x), or here, Q(2*12), that is, Q(24). Mother non-composite numeral comprises Q(2x−1), here, Q(23). As demonstrated by the binary tree file hierarchy embodiment illustrated in FIG. 4, this technique provides a file hierarchy of named file locations that is well-known to be useful for searching and/or sorting. See, for example, The Art of Computer Programming in Vol. 2: Searching and Sorting, by Donald K. Knuth, 3^(rd) Edition, 1997, available from Addison Wesley Longman. Thus, the table of FIG. 3 provides an infinite number of binary trees, each binary tree associated with a finite multi-set associated with two or more natural numerals larger than one.

The table of FIG. 3 also provides another approach to moving between named files in a file structure or file hierarchy. For this particular embodiment, this is accomplished by associating at least some named files in the file structure with products of non-composite numerals, and associating other named files in the file structure with non-composite numerals. As shall be explained in more detail hereinafter, in such an embodiment, moving between named files in a file hierarchy may be accomplished via a mathematical operation, such as by dividing and/or multiplying with non-composite numerals.

As previously described and illustrated by the tree of FIG. 1, in a Unix-based operating system, it may be time consuming in a relative sense to move between files in a file structure, using serial access. FIG. 5 illustrates an embodiment of an alternate file hierarchy in which moving between named files may accomplished through mathematical manipulation, thus, potentially saving time. In FIG. 5, the nodes represent named files that have particular representations in memory. For this example, let file 510 have associated with it the non-composite numeral Q(2x) This is analogous or equivalent to root node 110 of FIG. 1. Emanating from file 510 are thus a potentially infinite number of named files, such as, for example, files 520 and 530. In this particular embodiment, these named files are located serially in memory based at least in part on a product of two non-composite numerals. Thus, for example, assume it is desirable to move between file 510 and file 520. This may be accomplished simply by dividing by the non-composite numeral Q(2y₁−1) and then multiplying by the non-composite numeral Q(2y_(k)−1). Thus, moving between these named file locations reduces to a mathematical calculation. It is noted that this is simply one particular embodiment of this approach and many other embodiments may be employed. For example, a non-composite numeral other than Q(2x) may be employed as the core and other named files associated with the core may be located in a manner other than as illustrated in FIG. 5. Thus, it is intended to cover all such embodiments regardless of the particular non-composite numerals employed to name and locate such files.

Furthermore, it will, of course, be appreciated that, for convenience, in a particular embodiment, a translation table between characters and non-composite numerals may also be employed. Thus, in a typical operation, a first file may have a named location represented as a sequence of characters and a table look-up may be employed to obtain its designation or named location in terms of non-composite numerals or products of non-composite numerals, such as an embodiment like the one previously described. Mathematical operations may then be applied to the non-composite numerals or the products of non-composite numerals, in such an embodiment, to locate the named file it is desired to locate and then a translation table may be employed to provide the characters associated with the resulting non-composite numeral or product of non-composite numerals. In this particular embodiment, therefore, the essence of manipulating files or moving between files in a file hierarchy is reduced to division of non-composite numerals and/or products of non-composite numerals. In general, therefore, a method of constructing a file structure may comprise associating at least some files with non-composite numerals. It is, therefore, intended to cover all embodiments of such a file hierarchy.

It will, of course, be understood that, although particular embodiments have just been described, the claimed subject matter is not limited in scope to a particular embodiment or implementation. For example, one embodiment may be in hardware, such as implemented to operate on a device or combination of devices, for example, whereas another embodiment may be in software. Likewise, an embodiment may be implemented in firmware, or as any combination of hardware, software, and/or firmware, for example. Likewise, although the claimed subject matter is not limited in scope in this respect, one embodiment may comprise one or more articles, such as a storage medium or storage media. This storage media, such as, one or more CD-ROMs and/or disks, for example, may have stored thereon instructions, that when executed by a system, such as a computer system, computing platform, or other system, for example, may result in an embodiment of a method in accordance with the claimed subject matter being executed, such as one of the embodiments previously described, for example. As one potential example, a computing platform may include one or more processing units or processors, one or more input/output devices, such as a display, a keyboard and/or a mouse, and/or one or more memories, such as static random access memory, dynamic random access memory, flash memory, and/or a hard drive. For example, a display may be employed to display one or more queries, such as those that may be interrelated, and or one or more tree expressions, although, again, the claimed subject matter is not limited in scope to this example.

In the preceding description, various aspects of the claimed subject matter have been described. For purposes of explanation, specific numbers, systems and/or configurations were set forth to provide a thorough understanding of the claimed subject matter. However, it should be apparent to one skilled in the art having the benefit of this disclosure that the claimed subject matter may be practiced without the specific details. In other instances, well-known features were omitted and/or simplified so as not to obscure the claimed subject matter. While certain features have been illustrated and/or described herein, many modifications, substitutions, changes and/or equivalents will now occur to those skilled in the art. It is, therefore, to be understood that the appended claims are intended to cover all such modifications and/or changes as fall within the true spirit of the claimed subject matter. 

1. An article comprising: a storage medium; said storage medium having stored thereon a file location naming hierarchy; wherein said hierarchy comprises at least a multi-set (MS) of a set of elements.
 2. The article of claim 1, wherein said MS of a set of elements comprises a MS of a MS of said set of elements.
 3. The article of claim 1, wherein said set of elements comprise natural numerals.
 4. The article of claim 1, wherein said hierarchy further comprises compound multi-sets of said set of elements.
 5. A method comprising: forming a file location naming hierarchy; wherein said hierarchy comprises at least a multi-set (MS) of a set of elements.
 6. The method of claim 5, wherein said MS of a set of elements comprises a MS of a MS of said set of elements.
 7. The method of claim 5, wherein said set of elements comprise natural numerals.
 8. The method of claim 5, wherein said hierarchy further comprises compound multi-sets of said set of elements.
 9. A method of arranging files in a file location naming hierarchy comprising: forming a multi-set from a set of elements; forming at least one more multi-set from said multi-set; wherein said hierarchy comprises the multi-sets.
 10. The method of claim 9, wherein said forming at least one more multi-set comprises recursively forming at least one compound multi-set.
 11. The method of claim 10, wherein said recursively forming at least one compound multi-set comprises recursively forming a plurality of compound multi-sets.
 12. An article comprising: a storage medium having stored thereon instructions that, when executed, result in execution of a method of arranging files in a file location naming hierarchy as follows: forming a multi-set from a set of elements; forming at least one more multi-set from said multi-set; wherein said hierarchy comprises the mufti-sets.
 13. The article of claim 13, wherein said instructions when executed further result in: said forming at least one more multi-set comprising recursively forming at least one compound multi-set.
 14. The article of claim 13, wherein said instructions, when executed, further result in: said recursively forming at least one compound multi-set comprising recursively forming a plurality of compound multi-sets.
 15. A method of associating a natural numeral with a multi-set comprising: for the numeral 2, assigning a unique multi-set; and for even composite numerals 2K, assigning the merger of the multi-set for 2 with the multi-set for K.
 16. The method of claim 15, and further comprising: for the numerals 0 and 1, assigning additional unique multi-sets: and for odd non-composite numerals, assigning the associated Kleene enumeration index.
 17. The method of claim 16, and further comprising: for composite numerals K*L, assigning the merger of the multi-set for K with the multi-set for L.
 18. The method of claim 16, wherein, for the numerals 0 and 1, assigning additional unique multi-sets comprises assigning the empty set and singleton zero, respectively.
 19. The method of claim 15, wherein, for the numeral 2, assigning a unique multi-set comprises assigning doubleton zero.
 20. An article comprising: a storage medium having stored thereon instructions that, when executed, result in a method of associating a natural numeral with a multi-set as follows: for the numeral 2, assigning a unique multi-set; and for even composite numerals 2K, assigning the merger of the multi-set for 2 with the multi-set for K.
 21. The article of claim 20, wherein said instructions, when executed, further result in: for the numerals 0 and 1, assigning additional unique multi-sets; and for odd non-composite numerals, assigning the associated Kleene enumeration index.
 22. The article of claim 21, wherein said instructions, when executed, further result in: for composite numerals, K*L, assigning the merger of the multi-set for K with the multi-set for L.
 23. The article of claim 21, wherein said instructions, when executed, further result in: for the numerals 0 and 1, assigning additional unique multi-sets comprising assigning the empty set and singleton zero, respectively.
 24. The article of claim 20, wherein said instructions, when executed, further result in: for the numeral 2, assigning a unique multi-set comprising assigning doubleton zero.
 25. A method of associating a natural numeral with a numeral pair comprising: forming said numeral pair so that it includes a Kleene enumeration index natural numeral corresponding to a non-composite numeral and another numeral that produces said natural numeral when multiplied with said non-composite numeral.
 26. The method of claim 25, wherein said natural numeral that comprises a non-composite numeral is designated by said another numeral of said numeral pair comprising the numeral
 1. 27. The method of claim 25, and further comprising: forming a finite multi-set from said numeral pair. 28 The method of claim 27, and further composing: manipulating said finite multi-set. 29 The method of claim 28, and further comprising: forming another finite multi-set from another numeral pair; wherein said manipulating said finite multi-set comprises: combining the finite multi-sets.
 30. The method of claim 29, wherein said combining the finite multi-sets comprises: multiplying the associated natural numerals.
 31. The method of claim 29, wherein said combining the finite multi-sets comprises: merging the associated numeral pairs.
 32. The method of claim 25, and further comprising: associating another natural numeral with another numeral pair; and multiplying the natural numerals by merging the associated numeral pairs.
 33. The method of claim 32, wherein said merging numeral pairs comprises: combining [a,b] with [c,d] as follows: [a, b*Q(c)”*d], in which “*” represents multiplication and where the function Q(c) is a function that provides the non-composite numeral associated with the index c, as for the Kleene enumeration.
 34. The method of claim 32, wherein said numeral pairs are represented as binary numerals, and said multiplying said natural numerals includes using the same or fewer bits than associated with binary representations of said natural numerals by merging the binary numeral pairs.
 35. The method of claim 25, wherein said natural numeral is represented by a binary numeral representation of said numeral pair.
 36. An article comprising: a storage medium having stored thereon instructions that, when executed, result in execution of a method of associating a natural numeral with a numeral pair as follows: forming said numeral pair so that it includes a Kleene enumeration index natural numeral corresponding to a non-composite numeral and another numeral that produces said natural numeral when multiplied with said non-composite numeral.
 37. The article of claim 36, wherein said instructions, when executed, further result in: said natural numeral that comprises a non-composite numeral being designated by said another numeral of said numeral pair comprising a numeral
 1. 38. The article of claim 36, wherein said instructions, when executed, further result in: forming a finite multi-set from said numeral pair.
 39. The article of claim 38, wherein said instructions, when executed, further result in: manipulating said finite multi-set.
 40. The article of claim 39, wherein said instructions, when executed, further result in: forming another finite multi-set from another numeral pair; wherein manipulating said finite multi-set comprises: combining the finite multi-sets.
 41. The article of claim 40, wherein said instructions, when executed, further result in: said combining the finite multi-sets comprising multiplying the associated natural numerals.
 42. The article of claim 40, wherein said instructions, when executed, further result in: said combining the finite multi-sets comprising merging the associated numeral pairs.
 43. The article of claim 36, wherein said instructions, when executed, further result in: associating another natural numeral with another numeral pair; and multiplying the natural numerals by merging the associated numeral pairs.
 44. The article of claim 43, wherein said instructions, when executed, further result in: said merging the associated numeral pairs comprising combining [a,b] with [c,d] as follows: [a, b*Q(c)”], in which “*” represents multiplication and where the function Q(c) is a function that provides the non-composite numeral associated with the index c, as for the Kleene enumeration.
 45. The article of claim 43, wherein said instructions, when executed, further result in: said numeral pairs being represented as binary numerals: and said multiplying said natural numerals induding using the same or fewer bits than associated with binary representations of said natural numerals by merging the binary numeral pairs.
 46. The article of claim 36, wherein said instructions, when executed, further result in: said natural numeral being represented by a binary numeral representation of said numeral pair.
 47. An article comprising: a storage medium having stored thereon a table; wherein said table uniquely associates a plurality of natural numerals with a plurality of numeral pairs, wherein at least one of the numerals for each of said pairs represents an index for the Kleene enumeration.
 48. The erode of claim 47, wherein said table further uniquely associates a plurality of finite multi-sets with said natural numerals and said numeral pairs.
 49. The article of claim 48, wherein, for the numerals 0, 1, and 2, the associated multi-sets comprise the empty set singleton zero and doubleton zero, respectively.
 50. The article of claim 49, wherein, for composite numerals 2K, the associated multi-set comprises a merger of singleton zero with the multi-set associated with K. 51 The article of claim 50, wherein, for composite numerals K*L, the associated multi-set comprises a merger of the multi-set associated with K with the multi-set associated with L.
 52. A method comprising: uniquely associating a plurality of natural numerals with a plurality of numeral pairs, wherein at least one of the numerals for each of said pairs represents an index for the Kleene enumeration.
 53. The method of claim 52, wherein uniquely associating further comprises uniquely associating a plurality of finite multi-sets with sad natural numerals and said numeral pairs.
 54. The method of claim 53, wherein uniquely associating, for the numerals 0, 1, and 2, comprises associating the multi-sets empty set, singleton zero and doubleton zero, respectively.
 55. The method of claim 54, wherein uniquely associating for composite numerals 2K, comprises associating the multi-set comprising a merger of singleton zero with the multi-set associated with K.
 56. The method of claim 55, wherein uniquely associating, for composite numerals K*L, comprises associating the multi-set comprising a merger of the multi-set associated with K with the multi-set associated with L.
 57. A method of forming a binary tree file location naming hierarchy from a finite multi-set comprising: associating said finite multi-set with a natural numeral; and associating a first non-composite numeral and a second non-composite numeral with said natural numeral, said non-composite numerals and said finite multi-set forming at least a portion of said hierarchy.
 58. The method of claim 57, wherein said finite multi-set represents a named location for a file in said hierarchy, and said first and second non-composite numerals represent additional named locations for related files in said hierarchy.
 59. The method of claim 57, and further comprising associating a plurality of binary trees with finite multi-sets, each binary tree associated with a finite multi-set further associated with two or more natural numerals larger than one.
 60. An article comprising: a storage medium having stored thereon instructions that, when executed, result in: a method of forming a binary tree file location naming hierarchy from a finite multi-set as follows: associating said finite multi-set with a natural numeral; and associating a first non-composite numeral and a second non-composite numeral with said natural numeral, said non-composite numerals and said finite multi-set forming at least a portion of said hierarchy.
 61. The article of claim 60, wherein said instructions, when executed, further result in: said finite multi-set representing a named location for a file in said hierarchy, and said first and second non-composite numerals representing additional named locations for related files in said hierarchy.
 62. The article of claim 60, wherein said instructions, when executed, further result in: associating a plurality of binary trees with finite multi-sets, each binary tree associated with a finite multi-set further associated with two or more natural numerals larger than one.
 63. A method comprising: associating at least some files of a file location naming hierarchy with non-composite numerals.
 64. The method of claim 63, wherein said associating comprises associating said at least some files in said file location naming hierarchy with non-composite numerals and/or products of non-composite numerals; and further comprising: moving between said at least some files including dividing and/or multiplying said non-composite numerals and/or associated products with non-composite numerals.
 65. The method of claim 64 wherein said associating comprises associating other files other than said at least some files with non-composite numerals.
 66. The method of claim 65, wherein said moving between said at least some files comprising dividing and/or multiplying with non-composite numerals.
 67. The method of claim 64, wherein said associating comprises associating said at least some files with non-composite numerals; and said moving between said at least some files comprising multiplying and/or dividing by a non-composite numeral.
 68. An article comprising: a storage medium having stored thereon instructions that, when executed results in execution of a method as follows: associating at least some files of a file location naming hierarchy with non-composite numerals.
 69. The article of claim 68, wherein said instructions when executed further result in: said associating comprising associating said at least some files in said file location naming hierarchy with non-composite numerals and/or products of non-composite numerals; and further result in: moving between said at least some files including dividing and/or multiplying said non-composite numerals and/or associated products with non-composite numerals.
 70. The article of claim 69, wherein said instructions when executed further result in: said associating comprising associating other files other than said at least some files with non-composite numerals.
 71. The article of claim 70, wherein said instructions when executed further result in: said moving between said at least some files comprising dividing and/or multiplying with non-composite numerals.
 72. The article of claim 69, wherein said instructions when executed further result in: said associating comprising associating said at least some files with non-composite numerals; and said moving between said at least some files comprising multiplying and/or dividing by a non-composite numeral.
 73. An apparatus comprising: a computing platform; said computing platform having stored thereon a file location naming hierarchy; wherein said hierarchy comprises at least a multi-set (MS) of a set of elements.
 74. The apparatus of claim 73, wherein said MS of a set of elements comprises a MS of a MS of said set of elements.
 75. The apparatus of claim 73, wherein said set of elements comprise natural numerals.
 76. The apparatus of claim 73, wherein said hierarchy further comprises compound multi-sets of said set of elements.
 77. An article comprising: a file having a named file location; said named location having been formed at least by: forming a file location naming hierarchy; wherein said hierarchy comprises at least a multi-set (MS) of a set of elements.
 78. The article of claim 77, wherein said file having a named location having further been formed at least by: said MS of a set of elements comprising a MS of a MS of said set of elements.
 79. The article of claim 77, wherein said file having a named location having further been formed at least by: said set of elements comprising natural numerals.
 80. The article of claim 77, wherein said file having a named location having further been formed at least by: said hierarchy comprising compound multi-sets of said set of elements.
 81. An article comprising: a file having a named location having been arranged in a file location naming hierarchy having been formed at least by: forming a multi-set from a set of elements; forming at least one more multi-set from said multi-set; said hierarchy comprising the multi-sets.
 82. The article of claim 81, wherein said named location having further been formed at least by forming at least one more multi-set comprising recursively forming at least one compound multi-set.
 83. The article of claim 82, wherein said named location having further been formed at least by said recursively forming at least one compound comprising recursively forming a plurality of compound multi-sets.
 84. An apparatus comprising: a computing platform; said computing platform adapted to form a multi-set from a set of elements; adapted to form at least one more multi-set from said multi-set; and adapted so that said hierarchy comprises the multi-sets.
 85. The apparatus of claim 84, wherein said computing platform is adapted to form at least one more multi-set by recursively forming at least one compound multi-set.
 86. The apparatus of claim 85, wherein said computing platform is adapted to recursively forming at least one compound multi-set by recursively forming a plurality of compound multi-sets.
 87. An article comprising: a file having a named location; said named location being formed from a multi-set associated with a natural numeral at least by: for 2, assigning a unique multi-set; and for even composite numerals, 2K, assigning the merger of the multi-set for 2 with the multi-set for K.
 88. The article of claim 87, said named location being formed from a multi-set further associated with a natural numeral at least by: for 0 and 1, assigning additional unique multi-sets; and for odd non-composite numerals, assigning the associated Kleene enumeration index.
 89. The article of claim 88, said named location being formed from a multi-set further associated with a natural numeral at least by: for composite numerals, K×L, assigning the merger of the multi-set for K with the multi-set for L.
 90. The article of claim 88, said named location being formed from a multi-set further associated with a natural numeral at least by: for 0 and 1, assigning unique multi-sets comprises assigning the empty and singleton zero, respectively.
 91. The article of claim 87, said named location being formed from a multi-set further associated with a natural numeral at least by: for 2, assigning a unique multi-set comprises assigning doubleton zero.
 92. An apparatus comprising: a computing platform; said computing platform being adapted to associate a natural numeral with a multi-set as follows: for 2, assigning a unique multi-set; and for even composite numerate, 2K, assigning the merger of the multi-set for 2 with the multi-set for K.
 93. The apparatus of claim 92, wherein said computing platform is further adapted to: for 0 and 1, assign additional unique multi-sets; and for odd non-composite numerals, assign the associated Kleene enumeration index.
 94. The apparatus of claim 93, wherein said computing platform is further adapted to: for composite numerals, K×L, assign the merger of the multi-set for K with the multi-set for L.
 95. The apparatus of claim 93, wherein said computing platform is further adapted to: for 0 and 1, assign unique multi-sets comprising the empty set and singleton zero, respectively.
 96. The apparatus of claim 92, wherein said computing platform is further adapted to: for 2, assign a unique multi-set comprising doubleton zero.
 97. An article comprising: a file having a named location; said named location being formed from a numerical pair associated with a natural numeral at least by: forming said numeral pair so that it includes an index natural numeral corresponding to a non-composite numeral and another numeral that produces said natural numeral when multiplied with said non-composite numeral.
 98. The article of claim 97, wherein said named location being formed from a numerical pair further associated with a natural numeral at least by: said natural numeral that comprises a non-composite numeral being designated by said another numeral of said numeral pair comprising a one numeral.
 99. The article of claim 97, wherein said named location being formed from a numerical pair further associated with a natural numeral at least by: forming a finite multi-set from said numeral pair.
 100. The article of claim 99, said named location being formed from a numerical pair further associated with a natural numeral at least by: manipulating said finite multi-set.
 101. The article of claim 100, said named location being formed from a numerical pair further associated with a natural numeral at least by: forming another finite multi-set from another numeral pair, wherein manipulating said finite multi-set comprises: combining said finite multi-sets.
 102. The article of claim 101, wherein said named location being formed from a numerical pair associated with a natural numeral at least by: said combining said finite multi-sets comprising multiplying the associated natural numerals.
 103. The article of claim 101, wherein said named location being formed from a numerical pair further associated with a natural numeral at least by; said combining said finite multi-sets comprising merging the associated numeral pairs.
 104. The article of claim 97, said named location being formed from a numerical pair further associated with a natural numeral at least by: associating another natural numeral with another numeral pair and multiplying natural numerals by merging associated numeral pairs.
 105. The article of claim 104, wherein said named location being formed from a numerical pair further associated with a natural numeral at least by said merging numeral pairs comprising combining [a,b] with [c,d] as follows: [a,b*Q(c)”*d], in which “*” represents multiplication and where the function Q(c) is a function that provides the non-composite numeral associated with the index c, as for the Kleene enumeration.
 106. The article of claim 104, wherein said named location being formed from a numerical pair further associated with a natural numeral at least by; said numeral pairs being represented as binary numerals, and multiplying said natural numerals using the same or fewer bits than associated with binary representations of said natural numerals by merging the binary numeral pairs.
 107. The article of claim 97, wherein sad named location being formed from a numerical pair further associated with a natural numeral at least by: said natural numeral being represented by a binary numeral representation of said numeral pair.
 108. An apparatus comprising: a computing platform; said computing platform being adapted to associate a natural numeral with a numeral pair as follows: forming said numeral pair so that it includes an index natural numeral corresponding to a non-composite numeral and another numeral that produces said natural numeral when multiplied with said non-composite numeral.
 109. The apparatus of claim 108, wherein said computing platform being further adapted to associate a natural numeral with a numeral pair by said natural numeral that comprises a non-composite numeral being designated by said another numeral of said numeral pair comprising a one numeral.
 110. The apparatus of claim 108, wherein said computing platform is further adapted to form a finite multi-set from said numeral pair.
 111. The apparatus of claim 110, wherein said computing platform is further adapted to manipulate said finite multi-set.
 112. The apparatus of claim 111, wherein said computing platform is further adapted to form another finite multi-set from another numeral pair wherein manipulating said finite mutt-set comprises combining said finite multi-sets.
 113. The apparatus of claim 112, wherein said computing platform is further adapted to combine said finite multi-sets by multiplying the associated natural numerals.
 114. The apparatus of claim 112, wherein said computing platform is further adapted to combine said finite multi-sets by merging the associated numeral pairs.
 115. The apparatus of claim 108, wherein said computing platform is further adapted to associate another natural numeral with another numeral pair; and to multiply the natural numerals by merging associated numeral pairs.
 116. The apparatus of claim 115, wherein said computing platform is further adapted to combine [a,b] with [c,d] as follows: [a, b*Q(c)”*d], in which “*” represents multiplication and where the function Q(c) is a function that provides the non-composite numeral associated with the index c, as for the Kleene enumeration.
 117. The apparatus of claim 115, wherein said computing platform is further adapted to represent said numeral pairs as binary numerals: and to multiply said natural numerals using the same or fewer bits than associated with binary representations of said natural numerals by merging the binary numeral pairs.
 118. The apparatus of claim 108, wherein said computing platform being adapted to represent said natural numeral by a binary numeral representation of said numeral pair.
 119. An apparatus comprising: a computing platform; said computing platform adapted to uniquely associate a plurality of natural numerals with a plurality of numeral pairs, wherein at least one of the numerals for each of said pairs represents an index for the Kleene enumeration.
 120. The apparatus of claim 119, wherein said computing platform is further adapted to uniquely associate a plurality of finite multi-sets with said natural numerals and said numeral pairs.
 121. The apparatus of claim 120, wherein said computing platform is further adapted to associate, for the numerals 0, 1, and 2, the multi-sets, the empty set, singleton zero and doubleton zero respectively.
 122. The apparatus of claim 121, wherein computing platform is further adapted to associated, for composite numerals, 2K, the multi-set comprising a merger of singleton zero with the multi-set associated with K.
 123. The apparatus of claim 122, wherein said computing platform is further adapted to associate, for composite numerals, K*L, the multi-set comprising a merger of the multi-set associated with K with the multi-set associated with L.
 124. An article comprising: a file; said file raving a named location: said named location being part of a file location naming hierarchy formed at least by: associating a plurality of natural numerals with a plurality of numeral pairs, wherein at least one of the numerals for each of said pairs represents an index for the Kleene enumeration.
 125. The article of claim 124, wherein said file location naming hierarchy is further formed by: uniquely associating a plurality of finite multi-sets with said natural numerals and said numeral pairs.
 126. The article of claim 125, wherein said file location naming hierarchy is further formed by: associating, for the numerals 0, 1, and 2, the multi-sets the empty set singleton zero and doublet on zero, respectively.
 127. The article of claim 126, wherein said file location naming hierarchy is further formed by: associating, for composite numerals 2K, the multi-set comprising a merger of singleton zero with the multi-set associated with K.
 128. The article of claim 127, wherein said file location naming hierarchy is further formed by: associating, for composite numerals K*L, the multi-set comprising a merger of the multi-set associated with K with the multi-set associated with L.
 129. An article comprising; a file; said file having a named location; said named location being part of a binary tree file location naming hierarchy at least formed from a finite mufti-set by: associating said finite multi-set with a natural numeral; and associating a first non-composite numeral and a second non-composite numeral with said natural numeral, said non-composite numerals and said finite multi-set forming at least a portion of said hierarchy.
 130. The article of claim 129, wherein said file location naming hierarchy is further formed by: said finite multi-set representing a named location for a file in said hierarchy, and said first and second non-composite numerals representing additional named locations for related files in said hierarchy.
 131. The article of claim 129, wherein said file location naming hierarchy is further formed by: associating a plurality of binary trees with finite multi-sets, each binary tree associated with a finite multi-set further associated with two or more natural numerals larger than one.
 132. An apparatus comprising: a computing platform; said computing platform adapted to form a binary tree file location naming hierarchy from a finite multi-set as follows: associating said finite multi-set with a natural numeral; associating a first non-composite numeral and a second non-composite numeral with said natural numeral, said non-composite numerals and said finite multi-set forming at least a portion of said hierarchy. 133 The apparatus of claim 132, wherein said computing platform being further adapted to represent a named location for a file m said hierarchy with said finite multi-set, and to represent additional named locations for related files in said hierarchy with said first and second non-composite numerals.
 134. The apparatus of claim 132, wherein sad computing platform is further adapted to associate a plurality of number of binary trees with finite multi-sets, each binary tree associated with a finite multi-set further associated with two or more natural numerals larger than one.
 135. An article comprising: a plurality of files; said files having a named location in a file location naming hierarchy; said hierarchy being formed at least by: associating at least some files of said file location naming hierarchy with non-composite numerals.
 136. The article of claim 135, wherein said hierarchy is further formed at least by: associating said at least some files in said file location naming hierarchy with non-composite numerals and/or products of non-composite numerals; and wherein at least some files in said hierarchy are moved between by: dividing and/or multiplying said non-composite numerals and/or associated products with non-composite numerals.
 137. The article of claim 136 wherein said hierarchy is further formed at least by: said associating comprising associating other files other than said at least some files with non-composite numerals.
 138. The article of claim 137, wherein at least some files in said hierarchy are moved between by: dividing and/or multiplying with non-composite numerals.
 139. The article of claim 136, wherein said hierarchy is further formed at least by: said associating comprising associating said at least some files with non-composite numerate; and wherein at least some files in said hierarchy are moved between by: multiplying and/or dividing by a non-composite numeral.
 140. An apparatus comprising: a computing platform; said computing platform being adapted to: associate at least some files of a file location naming hierarchy with non-composite numerals.
 141. The apparatus of claim 140, wherein said computing platform is further adapted to associate said at least some files in said file location naming hierarchy with non-composite numerals and/or products of non-composite numerals; and is further adapted to move between said at least some files by dividing and/or multiplying said non-composite numerals and/or associated products with non-composite numerals.
 142. The apparatus of claim 141, said computing platform is further adapted to associate other files other than said at least some files with non-composite numerals.
 143. The apparatus of claim 142, said computing platform is further adapted to move between said at least some files by dividing and/or multiplying with non-composite numerals.
 144. The apparatus of claim 141, wherein said computing platform is further adapted to associate said at least some files with non-composite numerals; and further adapted to movie between said at least some files by multiplying and/or dividing by a non-composite numerals.
 145. An apparatus comprising: means for forming a file location naming hierarchy; wherein said hierarchy comprises at least a multi-set (MS) of a set of elements; and means for accessing a file in said hierarchy.
 146. The apparatus of claim 145, wherein said MS of a set of elements comprises a MS of a MS of said set of elements.
 147. The apparatus of claim 145, wherein said set of elements comprise natural numerals.
 148. The apparatus of claim 145, wherein said hierarchy further comprises compound multi-sets of said set of elements.
 149. An apparatus comprising: means for arranging files in a file location naming hierarchy comprising: means for forming a multi-set from a set of elements; and means for forming at least one more multi-set from said multi-set; wherein said hierarchy comprises the multi-sets.
 150. The apparatus of claim 149, wherein said means for forming at least one more multi-set comprising means for recursively forming at least one compound multi-set.
 151. The apparatus of claim 150, wherein said means for recursively forming at least one compound multi-set comprising means for recursively forming a plurality of compound multi-sets.
 152. An apparatus comprising: means for associating a natural numeral with a multi-set comprising: means for assigning a unique multi-set, for the numeral 2; and means for assigning the merger of the multi-set for 2 with the multi-set for K, for even composite numerals 2K.
 153. The apparatus of claim 152, and further comprising: means for assigning additional unique multi-sets, for the numerals 0 and 1; and means for assigning the associated Kleene enumeration index, for odd non-composite numerals.
 154. The apparatus of claim 153, and further comprising: means for assigning the merger of the multi-set for K with the multi-set for L, for composite numerals K*L.
 155. The apparatus of claim 153, wherein said means for assigning additional unique multi-sets comprising means for assigning the empty set and singleton zero, for the numerals 0 and 1, respectively.
 156. The apparatus of claim 152, wherein said means for assigning a unique multi-set comprising means for assigning doubleton zero, for the numeral
 2. 157. An apparatus comprising: means for associating a natural numeral with a numeral pair comprising: means for forming said numeral pain; said means for forming said numeral pair comprising: means for including a Kleene enumeration index natural numeral corresponding to a non-composite numeral and means for including another numeral that produces said natural numeral when multiplied with said non-composite numeral.
 158. The apparatus of claim 157, and further comprising: means for designating said natural numeral that comprises a non-composite numeral; said means for designating comprising means for said another numeral of said numeral pair including a numeral
 1. 159. The apparatus of claim 157, and further comprising: means for forming a finite multi-set from said numeral pair.
 160. The apparatus of claim 159, and further comprising: means for manipulating said finite multi-set.
 161. The apparatus of claim 160, and further comprising: means for forming another finite multi-set from another numeral pair; wherein said means for manipulating said finite multi-set comprises: means for combining the finite multi-sets.
 162. The apparatus of claim 161, wherein said means for combining the finite multi-sets comprising: means for multiplying the associated natural numerals.
 163. The apparatus of claim 161, wherein said means for combining the finite multi-sets comprising: means for merging the associated numeral pairs.
 164. The apparatus of claim 157, and further comprising: means for associating another natural numeral with another numeral pair; and means multiplying the natural numerals comprising means for merging the associated numeral pairs.
 165. The apparatus of claim 164, wherein said means for merging the associated numeral pairs comprising, means for combining [a,b] with [c,d]; wherein said means for combining comprises [a, b*Q(c)”*d], which “*” represents multiplication and where the function Q(c) is a function that provides the non-composite numeral associated with the index c, as for the Kleene enumeration.
 166. The apparatus of claim 164, and further comprising: means for representing said numeral pairs as binary numerate; and wherein said means for multiplying said natural numerals including means for merging the binary numeral pairs using the same or fewer bits than associated with binary representations of said natural numerals.
 167. The apparatus of claim 157, and further comprising; means for representing said natural numeral with a binary numeral representation of said numeral pair.
 168. An apparatus comprising: means for uniquely associating a plurality of natural numerals with a plurality of numeral pairs, wherein at least one of the numerals for each of said pairs represents an index for the Kleene enumeration.
 169. The apparatus of claim 168, wherein said means for uniquely associating comprises means for uniquely associating a plurality of finite multi-sets with said natural numerals and said numeral pairs.
 170. The apparatus of claim 169, and further comprising: means for uniquely associating, for the numerals 0, 1, and 2, the multi-sets comprising the empty set, singleton zero and doubleton zero, respectively.
 171. The apparatus of claim 170, and further comprising: means for uniquely associating, for composite numerals 2K, the multi-set comprising a merger of singleton zero with the multi-set associated with K. 172 The apparatus of claim 171, and further comprising: means for uniquely associating, for composite numerals K*L, the multi-set comprising a merger of the multi-set associated with K with the multi-set associated with L.
 173. An apparatus comprising: means for forming a binary tree file location naming hierarchy from a finite multi-set comprising: means for associating said finite multi-set with a natural numeral; and means for associating a first non-composite numeral and a second non-composite numeral with said natural numeral, said non-composite numerals and said finite multi-set forming at least a portion of said hierarchy.
 174. The apparatus of claim 173, and further comprising: means for representing a named location for a file in said hierarchy with said finite multi-set, and means for representing additional named locations for related files in said hierarchy with said first and second non-composite numerals.
 175. The apparatus of claim 173, and further comprising: means for associating a plurality of binary trees with finite multi-sets, each binary tree associated with a finite multi-set further associated with two or more natural numerals larger than one.
 176. An apparatus comprising means for associating at least some files of a file location naming hierarchy with non-composite numerals.
 177. The apparatus of claim 176, wherein said means for associating comprising means for associating said at least some files in said file location naming hierarchy with non-composite numerals and/or products of non-composite numerals, and further comprising means for moving between said at least some lies including dividing and/or multiplying said non-composite numerals and/or associated products with non-composite numerals.
 178. The apparatus of claim 177, wherein said means for associating comprising means for associating other files other than said at least some files with non-composite numerals.
 179. The apparatus of claim 178, said means for moving between said at least some files comprising dividing and/or multiplying with non-composite numerals.
 180. The apparatus of claim 177, wherein said means for associating comprising means for associating said at least some files with non-composite numerals; and said means for moving between said at least some files comprising multiplying and/or dividing by a non-composite numeral. 